Paramagnetic phases of Kagome lattice quantum Ising models p.1/16

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1 Paramagnetic phases of Kagome lattice quantum Ising models Predrag Nikolić In collaboration with T. Senthil Massachusetts Institute of Technology Paramagnetic phases of Kagome lattice quantum Ising models p.1/16

2 Overview Introduction Quantum Ising models Motivation Transverse field Ising model on the Kagome lattice Dynamics of individual spins Disordered phase for all strengths of transverse field XXZ model on the Kagome lattice Dynamics of frustrated bonds Disordered, spin liquid and valence-bond ordered phases Paramagnetic phases of Kagome lattice quantum Ising models p.2/16

3 Quantum Paramagnetic Phases Singlet valence-bond: 2 staggered VBC plaquette VBC spin liquid (RVB) Promising spin systems: Checkerboard Kagome Pyrochlore Paramagnetic phases of Kagome lattice quantum Ising models p.3/16

4 Models Transverse field quantum Ising model (TFIM) H = J z S z i S z j Γ S i x, Γ J z i j i XXZ model: total Ising spin conserved H = J z S z i S z j + J ( S x i S x j + S y i S y j), J J z i j i j S = 1 2 Nearest-neighbor Ising interaction Further-neighbor and multiple-spin exchange dynamics What phases are possible? Paramagnetic phases of Kagome lattice quantum Ising models p.4/16

5 Motivation for Kagome Ising Models Kagome Heisenberg a.f. Seemingly gapless modes in absence of continuous symmetry breaking Easy-axis anisotropy = XXZ model Kagome Ising a.f. in weak transverse fields Disordered ground-state Ch. Waldtmann H.-U. Everts B. Bernu P. Sindzingre C. Lhuillier P. Lecheminant L. Pierre R. Moessner S. L. Sondhi Search for unconventional quantum phases Conditions in which various phases occur Kagome phases: disordered, spin liquid, VBC Paramagnetic phases of Kagome lattice quantum Ising models p.5/16

6 U(1) Gauge Theory: Spin Picture H z = J i j S z i S z j = J 2 ( i S z i) 2 + const. Minimum frustration = local constraint ( p ) s z p = S z pq { ± 1 } 2 q p U(1) gauge theory on honeycomb lattice Electric field vector E pq Charged boson n p Mapping lattice quantity condition Kagome site i triangle p S z i s z p min. frustration honeycomb bond pq site p E pq n p Gauss Law Paramagnetic phases of Kagome lattice quantum Ising models p.6/16

7 Analysis 2D compact U(1) gauge theory with bosonic matter field = a non-topological disordered phase exists (in 2D) 1. Duality transformation = integer-valued gauge theory on triangular lattice study dynamics of vortices 2. Relax integer-constraints = sine-gordon theory 3. Integrate out high-energy fields = effective XY model with 3-state anisotropy 4. Find continuum limit = field theory, explore the phase diagram Paramagnetic phases of Kagome lattice quantum Ising models p.7/16

8 Phases: Transverse Field Ising Model Disordered non-topological phase (Higgs) agrees with Monte-Carlo (Moessner, Sondhi) Valence-bond ordered phase broken translational symmetry (3-fold degeneracy) broken global Z 2 symmetry < M > 1 7/9 5/9 1/3 1/ h/j Dominant hexagon ring exchange... Ordered phase is perhaps related to the plateau in magnetization curve Paramagnetic phases of Kagome lattice quantum Ising models p.8/16

9 U(1) Gauge Theory: Bond Picture Frustrated bond dimer Minimal frustration: one dimer per Kagome triangle many degenerate ground-states Quantum fluctuations lift degeneracy How: Quantum dimer model on the dice lattice Order-by-disorder? entropically selected ordered state? Paramagnetic phases of Kagome lattice quantum Ising models p.9/16

10 Outline of Calculations Dice lattice is bipartite quantum dimer model compact U(1) gauge theory Dimers are soft-core bosonic matter field in the gauge theory distinguishes Kagome from other 2D frustrated systems disordered phases are possible Duality transformation lattice field theory ( height model) Exploring the phase diagram field-theoretical methods: extended mean-field Paramagnetic phases of Kagome lattice quantum Ising models p.10/16

11 Extended Mean-Field Method Mean-field state is determined by minimum of energy and maximum of entropy = minimize free energy Take into account effects of quantum fluctuations = seek order-by-disorder Formalism: Consider a path-integral with action S (Φ) Calculate free energy F(Φ 0 ) of a microstate Φ 0 : e F(Φ 0) = DΦ e S (Φ) m2 d d ( ) 2 Φ( ) Φ 0 ( ) Find the microstates Φ 0 that minimize F(Φ 0 ) Paramagnetic phases of Kagome lattice quantum Ising models p.11/16

12 Transverse Field Ising Model Dimer flips consistent with minimal frustration: Result: disorder-by-disorder entropically selected states are macroscopically numerous and generally disordered No phase transitions as Γ Agrees with Monte-Carlo: R. Moessner, S. L. Sondhi; Phys. Rev. B 63, (2001) Different theory, same result: U(1) gauge theory on the honeycomb lattice Paramagnetic phases of Kagome lattice quantum Ising models p.12/16

13 XXZ and Spin-Conserving Models More complicated dimer dynamics: (1) (2) (3) (4) Result: two possible phases valence-bond crystal (XXZ) spin liquid (multiple-spin exchange... ) Any phase with no broken lattice symmetries and conserved total Ising moment has topological order. Paramagnetic phases of Kagome lattice quantum Ising models p.13/16

14 XXZ Variational States H = J z S z i S z j ± J ( S x i S x j + S y i S y j), 0 < J J z i j i j J z : Minimize Ising frustration J : Maximize the number of valence-bonds: J z : Avoid frustration on the defect triangles: J : Minimize the cost of defect triangles by maximizing the number of perfect hexagons Paramagnetic phases of Kagome lattice quantum Ising models p.14/16

15 XXZ Variational States Visual overlap between the entropically selected state and variational states: Valence-bond order proposed for the Heisenberg model P. Nikolic, T. Senthil; Phys. Rev. B 68, (2003) Paramagnetic phases of Kagome lattice quantum Ising models p.15/16

16 Conclusions: Kagome Ising Phases type of dominant simple multiple-spin and Ising dynamics: short-ranged ring exchange... does not conserve i conserves i S z i S z i hexagon ring-exchange disordered disordered (TFIM) valence-bond spin liquid crystal (XXZ) magnetized valence-bond crystal Valence-bond order proposed for the Heisenberg model Found in the effective Z 2 gauge theory of the gapless singlets Paramagnetic phases of Kagome lattice quantum Ising models p.16/16